In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is numerical equality. Any number is equal to itself (reflexive). If , then (symmetric). If and , then (transitive).
Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
Notation
Various notations are used in the literature to denote that two elements and of a set are equivalent with respect to an equivalence relation the most common are "" and "", which are used when is implicit, and variations of "", "", or "" to specify explicitly. Non-equivalence may be written "" or "".
Definitions
A binary relation on a set is said to be an equivalence relation if it is reflexive, symmetric and transitive. That is, for all and in (reflexivity). if and only if (symmetry). If and then (transitivity).
together with the relation is called a setoid. The equivalence class of under denoted is defined as
Alternative definition using relational algebra
In relational algebra, if and are relations, then the composite relation is defined so that if and only if there is a such that and .Sometimes the composition is instead written as , or as ; in both cases, is the first relation that is applied. See the article on Composition of relations for more information. This definition is a generalisation of the definition of functional composition. The defining properties of an equivalence relation on a set can then be reformulated as follows: . (reflexivity). (Here, denotes the identity function on .) (symmetry). (transitivity).
Examples
Simple example
On the set , the relation is an equivalence relation. The following sets are equivalence classes of this relation:
The set of all equivalence classes for is This set is a partition of the set . It is also called the quotient set of by .
Equivalence relations
The following relations are all equivalence relations: "Is equal to" on the set of numbers. For example, is equal to "Is similar to" on the set of all triangles. "Is congruent to" on the set of all triangles. Given a function , "has the same image under as" on the elements of 's domain . For example, and have the same image under , viz. . In particular: "Has the same absolute value as" on the set of real numbers "Has the same cosine as" on the set of all angles. Given a natural number , "is congruent to, modulo " on the integers. "Have the same length and direction" (equipollence) on the set of directed line segments.Lena L. Severance (1930) The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis, link from HathiTrust "Has the same birthday as" on the set of all people.
Relations that are not equivalences
The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7. The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty). The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point, then this defines an equivalence relation.
Connections to other relations
A partial order is a relation that is reflexive, , and transitive. Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. A strict partial order is irreflexive, transitive, and asymmetric. A partial equivalence relation is transitive and symmetric. Such a relation is reflexive if and only if it is total, that is, if for all there exists some If: Given let hold using totality, then by symmetry, hence by transitivity. — Only if: Given choose then by reflexivity. Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and total relation. A ternary equivalence relation is a ternary analogue to the usual (binary) equivalence relation. A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. A preorder is reflexive and transitive. A congruence relation is an equivalence relation whose domain is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the normal subgroups). Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle. Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.
Well-definedness under an equivalence relation
If is an equivalence relation on and is a property of elements of such that whenever is true if is true, then the property is said to be well-defined or a under the relation
A frequent particular case occurs when is a function from to another set if implies then is said to be a for a or simply This occurs, e.g. in the character theory of finite groups. The latter case with the function can be expressed by a commutative triangle. See also invariant. Some authors use "compatible with " or just "respects " instead of "invariant under ".
More generally, a function may map equivalent arguments (under an equivalence relation ) to equivalent values (under an equivalence relation ). Such a function is known as a morphism from to
Related important definitions
Let , and be an equivalence relation. Some key definitions and terminology follow:
Equivalence class
A subset of such that holds for all and in , and never for in and outside , is called an equivalence class of by . Let denote the equivalence class to which belongs. All elements of equivalent to each other are also elements of the same equivalence class.
Quotient set
The set of all equivalence classes of by denoted is the quotient set of by If is a topological space, there is a natural way of transforming into a topological space; see Quotient space for the details.
Projection
The projection of is the function defined by which maps elements of into their respective equivalence classes by Theorem on projections:Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea. Let the function be such that if then Then there is a unique function such that If is a surjection and then is a bijection.
Equivalence kernel The equivalence kernel of a function is the equivalence relation ~ defined by The equivalence kernel of an injection is the identity relation.
Partition
A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single element of P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X.
Counting partitions
Let X be a finite set with n elements. Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number Bn: (Dobinski's formula).
Fundamental theorem of equivalence relations
A key result links equivalence relations and partitions:Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 29–32, Marcel Dekker An equivalence relation ~ on a set X partitions X. Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X. In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X.
Comparing equivalence relations
If and are two equivalence relations on the same set , and implies for all then is said to be a coarser relation than , and is a finer relation than . Equivalently, is finer than if every equivalence class of is a subset of an equivalence class of , and thus every equivalence class of is a union of equivalence classes of . is finer than if the partition created by is a refinement of the partition created by .
The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest.
The relation " is finer than " on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice.. Sect. IV.9, Theorem 12, page 95
Generating equivalence relations
Given any set an equivalence relation over the set of all functions can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixed points have the same cardinality. An equivalence relation on is the equivalence kernel of its surjective projection Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea. Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain. Thus an equivalence relation over a partition of and a projection whose domain is are three equivalent ways of specifying the same thing. The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of ) is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation is the intersection of all equivalence relations containing R (also known as the smallest equivalence relation containing R). Concretely, R generates the equivalence relation if there exists a natural number and elements such that , , and or , for The equivalence relation generated in this manner can be trivial. For instance, the equivalence relation generated by any total order on X has exactly one equivalence class, X itself.